Prove that if $n>1$, the sum of positive integers less than $n$ and coprime to $n$ is $(1/2)na(n)$ where $a(n)$ is the number of such integers. [duplicate]

Question 12(iii) Could anyone explain this part of the question to me.

What i tried co-prime means that the two integers a and b are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1

Take th number $$3$$ for example, then the sum of integers less than $$3$$ and co-prime to $$3$$ is $$2+1=3$$, $$2$$ and $$1$$ are the two integers co-prime to $$3$$ which thus satisfies the formula $$0.5*n*a(n)$$ where $$n=3$$, $$a(n)=2$$. However im unsure of how to prove it. Could anyone explain. Thanks

Hint

You have already shown that $$(m,n)=1 \iff (m,m-n)=1$$

See if you can figure out what is going on with the sum of such integers.

By the way, the notation for the Euler function is $$\phi (n)$$ not $$\alpha (n)$$

Hint:

If $$m$$ is coprime to $$n$$ then $$n-m$$ is also coprime to $$n$$.

• Isn't thats what was given in the first part of the question. – Nicole Alison Oct 12 '18 at 18:59